Composition of continuous maps is continuous

Statement

Let $X$, $Y$ and $Z$ be topological spaces. Let $f \colon X \to Y$ and $g \colon Y \to Z$ be continuous maps. Then the composition $g \circ f \colon X \to Z$ is continuous.

Remarks

Proof: for each open set $W$ of $Z$, $(g \circ f)^{-1}(W) = f^{-1}(g^{-1}(W))$. The set $g^{-1}(W)$ is open in $Y$, so its preimage under $f$ is open in $X$. For example, $x \mapsto \sin(x^2)$ is continuous on $\mathbb{R}$, because $x \mapsto x^2$ and $\sin$ are continuous.

Explanations

L1 human

If you do one continuous operation and then another one, the result is continuous. Small changes in the input give small changes after the first step, and then small changes after the second step.